| Date | May 2016 | Marks available | 5 | Reference code | 16M.1.hl.TZ0.7 |
| Level | HL only | Paper | 1 | Time zone | TZ0 |
| Command term | Prove that | Question number | 7 | Adapted from | N/A |
Question
The function \(f:\mathbb{R} \to \mathbb{R}\) is defined by \(f:x \to \left\{ {\begin{array}{*{20}{c}} { - 3x + 1}&{{\text{for }}x < 0} \\ 1&{{\text{for }}x = 0} \\ {2{x^2} - 3x + 1}&{{\text{for }}x > 0} \end{array}} \right.\).
By considering limits prove that \(f\) is
continuous at \(x = 0\);
differentiable at \(x = 0\).
Markscheme
consider \(\mathop {\lim }\limits_{h \to 0 + } f(0 + h) = \mathop {\lim }\limits_{h \to 0 + } (2{h^2} - 3h + 1)\) M1
\( = 1 = f(0)\) A1
\(\mathop {\lim }\limits_{h \to 0 - } f(0 + h) = \mathop {\lim }\limits_{h \to 0 - } ( - 3h + 1)\) M1
\( = 1 = f(0)\) A1
hence \(f\) is continuous at \(x = 0\) AG
Note: The \( = f(0)\) needs only to be seen once.
[4 marks]
consider
\(\mathop {\lim }\limits_{h \to 0 + } \left( {\frac{{f(0 + h) - f(0)}}{h}} \right) = \mathop {\lim }\limits_{h \to 0 + } \left( {\frac{{2{h^2} - 3h + 1 - 1}}{h}} \right)\) M1A1
\( = \mathop {\lim }\limits_{h \to 0 + } \left( {\frac{{2{h^2} - 3h}}{h}} \right) = - 3\) A1
\(\mathop {\lim }\limits_{h \to 0 - } \frac{{f(0 + h) - f(0)}}{h} = \mathop {\lim }\limits_{h \to 0 - } \frac{{ - 3h + 1 - 1}}{h} = - 3\) M1A1
hence \(f\) is differentiable at \(x = 0\) AG
[5 marks]
Examiners report
Again this was a reasonably successful question for many candidates with full marks often being awarded. However a significant minority were let down by giving very informal and descriptive answers which only gained partial marks. As the command term in the question is “prove” there is a need for a degree of formality and an explicit use of limits was expected.
Again this was a reasonably successful question for many candidates with full marks often being awarded. However a significant minority were let down by giving very informal and descriptive answers which only gained partial marks. As the command term in the question is “prove” there is a need for a degree of formality and an explicit use of limits was expected.
